What is the general solution for 7tan(4x+199)=8sin(4x+199) ?

The general solution for 7tan(4x+199)=8sin(4x+199) is

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7tan(4x+199)=8sin(4x+199)

7 tan (4 x + 199) = 8 sin (4 x + 199)

x = \frac{πn}{2} - \frac{199}{4}, x = \frac{π}{4} - \frac{199}{4} + \frac{πn}{2}, x = - \frac{199}{4} + \frac{πn}{2} + \frac{0.50536 \dots}{4}, x = \frac{π}{2} - \frac{199}{4} + \frac{πn}{2} - \frac{0.50536 \dots}{4}

Solution steps

7 tan (4 x + 199) = 8 sin (4 x + 199)

Subtract

8 sin (4 x + 199)

from both sides

7 tan (4 x + 199) - 8 sin (4 x + 199) = 0

Express with sin, cos

\frac{7 sin ( 199 + 4 x ) - 8 cos ( 199 + 4 x ) sin ( 199 + 4 x )}{cos ( 199 + 4 x )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

7 sin (199 + 4 x) - 8 cos (199 + 4 x) sin (199 + 4 x) = 0

Factor

7 sin (199 + 4 x) - 8 cos (199 + 4 x) sin (199 + 4 x) : - sin (199 + 4 x) (8 cos (199 + 4 x) - 7)

- sin (199 + 4 x) (8 cos (199 + 4 x) - 7) = 0

Solving each part separately

sin (199 + 4 x) = 0 or 8 cos (199 + 4 x) - 7 = 0

sin (199 + 4 x) = 0 : x = \frac{πn}{2} - \frac{199}{4}, x = \frac{π}{4} - \frac{199}{4} + \frac{πn}{2}

8 cos (199 + 4 x) - 7 = 0 : x = - \frac{199}{4} + \frac{πn}{2} + \frac{arccos ( \frac{7}{8} )}{4}, x = \frac{π}{2} - \frac{199}{4} + \frac{πn}{2} - \frac{arccos ( \frac{7}{8} )}{4}

Combine all the solutions

x = \frac{πn}{2} - \frac{199}{4}, x = \frac{π}{4} - \frac{199}{4} + \frac{πn}{2}, x = - \frac{199}{4} + \frac{πn}{2} + \frac{arccos ( \frac{7}{8} )}{4}, x = \frac{π}{2} - \frac{199}{4} + \frac{πn}{2} - \frac{arccos ( \frac{7}{8} )}{4}

Show solutions in decimal form

x = \frac{πn}{2} - \frac{199}{4}, x = \frac{π}{4} - \frac{199}{4} + \frac{πn}{2}, x = - \frac{199}{4} + \frac{πn}{2} + \frac{0.50536 \dots}{4}, x = \frac{π}{2} - \frac{199}{4} + \frac{πn}{2} - \frac{0.50536 \dots}{4}

4 sin^{2} (x) + 4 cos (x) - 5 = 0, 0 \leq x \leq 2 π

2 sin (2 θ) = 1

cos (θ) = \frac{1}{2}, 0 \leq θ \leq 4 π

sin (x) a = \frac{π}{2}

tan^{2} (3 θ) = 1

What is the general solution for 7tan(4x+199)=8sin(4x+199) ?
The general solution for 7tan(4x+199)=8sin(4x+199) is